The continuous uniform distribution, or rectangular distribution, is a family of symmetric probability distributions such that for each member of the family all intervals of the same length on the distribution's support are equally probable. The support is defined by the two parameters, $a$ and $b$, which are its minimum and maximum values. The distribution is often abbreviated $U(a, b)$. It is the maximum entropy probability distribution for a random variate $X$ under no constraint other than that it is contained in the distribution's support.

The probability that a uniformly distributed random variable falls within any interval of fixed length is independent of the location of the interval itself (but it is dependent on the interval size), so long as the interval is contained in the distribution's support.

To see this, if $X \sim U(a, b)$ and $[x, x+d]$ is a subinterval of $[a, b]$ with fixed $d>0$, then, the formula shown:

$\displaystyle {f(x) = \begin{cases} \frac { 1 }{ b-a } &\text{for } a\le x\le b \\ 0 & \text{if } x \; \text{<} \; a \; \text{or} \; x \; \text{>} \; b \end{cases}}$

Is independent of $x$. This fact motivates the distribution's name.

Applications of the Uniform Distribution

When a $p$-value is used as a test statistic for a simple null hypothesis, and the distribution of the test statistic is continuous, then the $p$-value is uniformly distributed between 0 and 1 if the null hypothesis is true. The $p$-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. One often "rejects the null hypothesis" when the $p$-value is less than the predetermined significance level, which is often 0.05 or 0.01, indicating that the observed result would be highly unlikely under the null hypothesis. Many common statistical tests, such as chi-squared tests or Student's $t$-test, produce test statistics which can be interpreted using $p$-values.

Sampling from a Uniform Distribution

There are many applications in which it is useful to run simulation experiments. Many programming languages have the ability to generate pseudo-random numbers which are effectively distributed according to the uniform distribution.

If $u$ is a value sampled from the standard uniform distribution, then the value $a+(b-a)u$ follows the uniform distribution parametrized by $a$ and $b$.

Sampling from an Arbitrary Distribution

The uniform distribution is useful for sampling from arbitrary distributions. A general method is the inverse transform sampling method, which uses the cumulative distribution function (CDF) of the target random variable. This method is very useful in theoretical work. Since simulations using this method require inverting the CDF of the target variable, alternative methods have been devised for the cases where the CDF is not known in closed form. One such method is rejection sampling.

The normal distribution is an important example where the inverse transform method is not efficient. However, there is an exact method, the Box–Muller transformation, which uses the inverse transform to convert two independent uniform random variables into two independent normally distributed random variables.

Example

Imagine that the amount of time, in minutes, that a person must wait for a bus is uniformly distributed between 0 and 15 minutes. What is the probability that a person waits fewer than 12.5 minutes?

Let $X$ be the number of minutes a person must wait for a bus. $a=0$ and $b=15$. $x \sim U(0, 15)$. The probability density function is written as:

$f(x) = \frac{1}{15} - 0 = \frac{1}{15}$ for $0 \leq x \leq 15$

We want to find $P(x<12.5)$.

The probability a person waits less than 12.5 minutes is 0.8333.

Catching a Bus

The Uniform Distribution can be used to calculate probability problems such as the probability of waiting for a bus for a certain amount of time.